Borel liftings of Borel sets: some decidable and undecidable statements
Memoirs of the American Mathematical Society
One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in ZF C. Our starting point is the following elementary, though non-trivial result: Consider X ⊂ 2 ω × 2 ω , set Y = π(X), where π denotes the canonical projection of 2 ω × 2 ω onto the first factor, and suppose that ( ) : "Any compact subset of Y is the projection of some compact subset of X". If moreover X is Π 0 2 then (see  or ) ( ) : "The restriction of π to some relatively
... some relatively closed subset of X is perfect onto Y "; it follows (see Section 0.2) that in the present case Y is also Π 0 2 . Notice that the reverse implication ( ) ⇒ ( ) holds trivially for any X and Y . But as we shall see the implication ( ) ⇒ ( ) for an arbitrary Borel set X ⊂ 2 ω × 2 ω is equivalent to the statement "∀α ∈ ω ω , ℵ 1 is inaccessible in L(α)". More precisely we shall prove that the validity of ( ) ⇒ ( ) for all X ∈ Σ 0 1+ξ+1 , is equivalent to "ℵ L ξ < ℵ 1 ". However we shall show independently, that when X is Borel one can, in ZF C, derive from ( ) the weaker conclusion that Y is also Borel and of the same Baire class as X. This last result solves an old problem about compact covering mappings (see Section 0.2). In fact these results are closely related to the following general boundedness principle Lift (X, Y ): "If any compact subset of Y admits a continuous lifting in X, then Y admits a continuous lifting in X", where by a lifting of Z ⊂ π(X) in X we mean a mapping on Z whose graph is contained in X. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of X and Y . We shall also prove a similar result for a variation of Lift (X, Y ) in which "continuous liftings" are replaced by "Borel liftings", and which answers a question of H. Friedman. Among other applications we obtain a complete solution to a problem which goes back to Lusin concerning the existence of Π 1 1 sets with all constituents in some given class Γ of Borel sets, improving earlier results by J. Stern and R. Sami. The proof of the main result will rely on a nontrivial representation of Borel sets (in ZF C) of a new type, involving a large amount of "abstract algebra". This representation was initially developed for the purposes of this proof, but has several other applications. is countable (in the universe). Since then, many other similar results were discovered in connection with various natural regularity properties. Staying inside projective classes of first level, which will be the setting of this work, let us mention the equivalence, due to Martin () and Harrington (), between Det(Σ 1 1 ) (determinacy of all Σ 1 1 games) and the existence of sharps for all reals. However, that similar phenomena might appear for classes of Borel sets is more surprising and was the starting point of this work. More precisely one can derive from the results of , ,  that there exists a regularity property A of pairs of sets of reals such that A(X, Y ) holds in ZF C whenever X is Π 0 2 or Y is Σ 0 2 and: and which is, like "measurability" or the "Perfect Set Theorem property", an "inner regularity property", in the sense that B(Y ) is of the form: "If all compact subsets of Y are small then Y is small" relatively to some smallness notion on sets of reals. Notice that since any projective set A is constructed from a Borel set B by taking projections and complements, one can present formally any property of A as a property of B (one can even assume that B is Π 0 2 ). But since a compact subset of A is not necessarily the projection of a compact subset of B, such artificial manipulations do not give rise to inner regular properties. Lightface version: It is also an empirical observation that, at least when dealing with projective classes of the first level, all equivalences for classical regularity properties of such a class Γ admit (in general by the same arguments) "parametrized effective" versions in which the "boldface" class Γ is replaced by Γ(α) the "lightface version with parameter α" (see paragraph 0.1 below). For example if we fix α ∈ ω ω then the Perfect Set Theorem holds for all Π 1 1 (α) sets, if and only if ℵ L(α) 1 is countable; and Det(Σ 1 1 (α)) is equivalent to .